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We study a free boundary problem modeling tumor growth with a T-periodicsupply $\Phi(t)$ of external nutrients. The model contains two parameters $\mu$and $\widetilde{\sigma}$. We first show that (i) zero radially symmetricsolution is globally stable if and only if $\widetilde{\sigma}\ge \frac{1}{T}\int_{0}^{T} \Phi(t) d t$; (ii) If $\widetilde{\sigma}<\frac{1}{T} \int_{0}^{T}\Phi(t) d t$, then there exists a unique radially symmetric positive solution$\left(\sigma_{*}(r, t), p_{*}(r, t), R_{*}(t)\right)$ with period $T$ and itis a global attractor of all positive radially symmetric solutions for all$\mu>0$. These results are a perfect answer to open problems in Bai and Xu[Pac. J. Appl. Math. 2013(5), 217-223]. Then, considering non-radiallysymmetric perturbations, we prove that there exists a constant $\mu_{\ast}>0$such that $\left(\sigma_{*}(r, t), p_{*}(r, t), R_{*}(t)\right)$ is linearlystable for $\mu<\mu_{\ast}$ and linearly unstable for $\mu>\mu_{\ast}$.

The existence and linear stability of periodic solution for a free  boundary problem modeling tumor growth with a periodic supply of external  nutrients